If $f_t$ is generic then $A$ if formed by a collection of curves in $S^2\times [0,1]$. So, $(P,0)$ is connected to either

1. $(P,1)$ --- this happens for the constant $f_t$.

2. $(-P,1)$ --- this happens for the projection of a generic sphere eversion, otherwise the orientation would not change.

3. $(-P,0)$ --- It can not happen --- this follow since the number of critical points counted with singes has to be equal $2$ for every $t$.

If $f_t$ is generic then $A$ is formed by a collection of curves in $S^2\times [0,1]$. So, $(P,0)$ is connected to either

1. $(P,1)$ --- this happens for the constant $f_t$.

2. $(-P,1)$ --- this happens for the projection of a generic sphere eversion, otherwise the orientation would not change.

3. $(-P,0)$ --- It can not happen --- this follow since the number of critical points counted with singes ("$+$" for min and max and "$-$" for saddle) has to be $2$ for every $t$.